equipped with the norm
kukZ = sup
0≤t≤Tku(t)kL∞+ sup
0<t≤T
(νt)1/2k∇u(t)kL∞ .
Proceeding as above one obtains the existence of a local solution u ∈Z of (2.14) for a slightly smaller value of T, which is determined by a condition of the form (2.16) where C0 is replaced by a larger constant.
Remark 2.5. Ifu0 6= 0, the local existence timeT given by the proof of Proposition 2.4 satisfies T = C1ν
ku0k2L∞
, (2.18)
whereC1 >0 is a universal constant. This implies in particular that, if we consider the maximal solution u ∈C0([0, T∗), X) of (2.14) in X, then either T∗ =∞, which means that the solution is global, orku(t)kL∞ → ∞ ast→T∗. More precisely, we must haveku(t)k2L∞ > C1ν(T∗−t)−1 for all t∈[0, T∗). Note also that estimate (1.10) follows from (2.15) and (2.18).
Remark 2.6. Using standard parabolic smoothing estimates, it is not difficult to show that, if u ∈ C0([0, T], X) is the mild solution of (2.1) constructed in Proposition 2.4, then u(x, t) is a smooth function for (x, t) ∈R2×(0, T] which satisfies the Navier-Stokes equations (2.1) in the classical sense, with the pressurep(x, t) given by (2.6), see [19, 20].
2.5 Global existence and a priori estimates
Take u0 ∈X such that divu0 = 0, and letu ∈C0([0, T∗), X) be the maximal solution of (2.1) with initial datau0, the existence of which follows from Proposition 2.4. In view of Remark 2.5, to prove that this solution is global (namely, T∗ = ∞), it is sufficient to show that the norm ku(t)kL∞ cannot blow up in finite time. The easiest way to do that is to consider the vorticity distributionω = curlu=∂1u2−∂2u1, which satisfies the advection-diffusion equation
∂tω+u· ∇ω = ν∆ω . (2.19)
We know from Proposition 2.4 that kω(t)kL∞ ≤ 2k∇u(t)kL∞ < ∞ for any t ∈ (0, T∗), hence shifting the origin of time we can assume without loss of generality thatω0 =ω(·,0)∈L∞(R2).
Now, the parabolic maximum principle [37] asserts that kω(t)kL∞ is a nonincreasing function of time, which gives the a priori estimate
kω(t)kL∞ ≤ kω0kL∞ , for all t≥0 . (2.20) Unfortunately, the bound (2.20) does not provide any direct control on ku(t)kL∞, because we work in an unbounded domain where the velocity field can have arbitrarily low frequencies. In Fourier space, the relation between ˆu=Fu and ˆω=Fω takes the simple form
ˆ
u(ξ) = −iξ⊥
|ξ|2 ω(ξ)ˆ , ξ ∈R2\ {0} , (2.21)
where ξ⊥ = (−ξ2, ξ1) if ξ = (ξ1, ξ2) ∈ R2. This shows that the first-order derivatives of the velocity field u satisfy
∂1u1 =−∂2u2 = R1R2ω , ∂1u2 = −R21ω , ∂2u1 = R22ω ,
whereR1, R2 are the Riesz transforms (2.7). In particular, we deduce the a priori estimate k∇u(t)kBMO ≤ Ckω(t)kL∞ ≤ Ckω0kL∞ , for allt≥0 . (2.22) We refer to Section 4.1 for a more detailed discussion of the Biot-Savart law inR2.
To go further we observe that, since divu= 0, we have the identity
(u· ∇)u = 12∇|u|2+u⊥ω , (2.23) where u⊥ = (−u2, u1) if u = (u1, u2). We can thus write the Navier-Stokes equations (2.1) in the equivalent form
∂tu+u⊥ω = ν∆u− ∇q , divu = 0 , (2.24) whereq =p+12|u|2. Applying the Leray-Hopf projection, we obtain the analog of (2.10)
∂tu+P(u⊥ω) = ν∆u , divu = 0. (2.25) Since ω is under control, the nonlinear term P(u⊥ω) can be considered as a linear expression in the velocity field u, and this strongly suggests that the solutions of (2.25) should not grow faster than exp(Ckω0kL∞t) as t → ∞. The problem with this naive argument is that we cannot estimate kP(u⊥ω)kL∞ in terms of kukL∞kωkL∞, because the Leray-Hopf projection P is not continuous on L∞(R2)2. This difficulty was solved in an elegant way by O. Sawada and Y. Taniuchi, who obtained the following result.
Proposition 2.7. [40] Assume that u0 ∈ X, divu0 = 0, and ω0 = curlu0 ∈ L∞(R2). Then the Navier-Stokes equations (2.1) have a unique global (mild) solution u ∈ C0([0,∞), X) with initial data u0. Moreover, we have the estimate
ku(t)kL∞ ≤ Cku0kL∞exp
CkωkL∞t
, t≥0 , (2.26)
for some universal constant C >0.
The proof of Proposition 2.7 relies on a clever Fourier-splitting argument which we now describe. Let ˆχ:R2→R be a smooth function such that
ˆ χ(ξ) =
(1 if |ξ| ≤1, 0 if |ξ| ≥2.
We further assume that χ is radially symmetric and nonincreasing along rays. Let χ = F−1χˆ be the inverse Fourier transform of ˆχ, so that χ ∈ S(R2). Given any δ > 0, we denote by Qδ the Fourier multiplier with symbol ˆχ(ξ/δ) :
(Qdδf)(ξ) = ˆχ(ξ/δ) ˆf(ξ) , ξ ∈R2 . (2.27) It is clear that Qδ is a bounded linear operator onS′(R2).
Lemma 2.8. There exists a constantC2 >0 such that the following bounds hold for anyδ >0.
1. kQδfkL∞ ≤C2kfkL∞, for any f ∈Cbu(R2);
2. kQδ∇PfkL∞ ≤C2δkfkL∞, for any f ∈Cbu(R2)2;
3. k(1−Qδ)ukL∞ ≤C2δ−1kωkL∞, for any u∈X withdivu = 0 and curlu=ω.
Proof. The first estimate follows immediately from Young’s inequality (see Section 4.4), because Qδ is the convolution operator with the integrable functionx7→δ2χ(δx), the L1 norm of which does not depend onδ. To prove the second estimate we have to show that, for anyj, k, ℓ∈ {1,2}, the Fourier multiplier M with symbol
m(ξ) = iξjξkξℓ
|ξ|2 χ(ξ/δ)ˆ , ξ∈R2\ {0} ,
is continuous on L∞(R2) with operator norm bounded byCδ. We observe that m(ξ) = δψ(ξ/δ)ˆ , where ψ(ξ) =ˆ iξjξkξℓ
|ξ|2 χ(ξ)ˆ .
It follows thatM f =ψδ∗f, whereψδ(x) =δ3ψ(δx) andψ=F−1ψ. It is clear thatˆ ψ∈C∞(R2) (because ˆψ has compact support), and from the explicit formula
ψ(x) = 1
2π∂j∂k∂ℓ Z
R2
log(|x−y|)χ(y) dy , x∈R2 ,
it is straightforward to verify that |ψ(x)| ≤ C|x|−3 for |x| ≥ 1. Thus ψ ∈ L1(R2), and using Young’s inequality we conclude that kM fkL∞ ≤ kψδkL1kfkL∞ = δkψkL1kfkL∞, which is the desired result.
Finally, to prove the third estimate in Lemma 2.8, we use formula (2.21) to derive the relation ˆ
u(ξ)−Qdδu(ξ) =
1−χ(ξ/δ)ˆ −iξ⊥
|ξ|2 ω(ξ) =ˆ 1
δφ(ξ/δ)ˆˆ ω(ξ) , where
φ(ξ) =ˆ
1−χ(ξ)ˆ −iξ⊥
|ξ|2 , ξ∈R2\ {0} . (2.28) As before, ifφ=F−1φ, this implies thatˆ k(1−Qδ)ukL∞ ≤δ−1kφkL1kωkL∞, so we only need to verify that φ∈L1(R2). Since ˆχ has compact support inR2, it follows from (2.28) that
φ(x) = 1 2π
x⊥
|x|2 + Φ(x) , x∈R2\ {0},
where Φ : R2 → R is smooth, hence φ is integrable on any bounded neighborhood of the origin. On the other hand, if we apply the Laplacian ∆ξ to both sides of (2.28), the resulting expression belongs to L2(R2,dξ). This shows that |x|2φ∈L2(R2,dx), henceφ is integrable on the complement of any neighborhood of the origin. Thus altogether φ∈ L1(R2), which is the desired result.
Proof of Proposition 2.7. Let u ∈ C0([0, T∗), X) be the maximal solution of (2.1) with initial datau0. Without loss of generality, we assume thatu0 6≡0, and we fixt∈(0, T∗). The idea is to control the low frequencies |ξ| ≤ 2δ in the solution u(t) using the integral equation (2.14), and the high frequencies |ξ| ≥ δ using the third estimate in Lemma 2.8 together with the a priori bound on the vorticity. The threshold frequencyδwill depend on time and on the solution itself.
Given anyδ >0, we apply the Fourier multiplierQδdefined in (2.27) to the integral equation (2.14) and obtain
Qδu(t) = S(νt)Qδu0− Z t
0
S(ν(t−s))Qδ∇ ·P(u(s)⊗u(s)) ds ,
where we have used the fact that Qδ commutes with the heat semigroup S(t). Using the first two estimates in Lemma 2.8, we thus find
kQδu(t)kL∞ ≤ C2ku0kL∞+C2δ Z t
0 ku(s)k2L∞ds . On the other hand, the third estimate in Lemma 2.8 implies that
k(1−Qδ)u(t)kL∞ ≤ C2δ−1kω(t)kL∞ ≤ C2δ−1kω0kL∞ .
This bound shows how the high frequencies in the velocity field u(t) can be controlled in terms of the vorticity. Combining both results, we find
ku(t)kL∞ ≤ C2ku0kL∞+C2δ Z t
0 ku(s)k2L∞ds+C2δ−1kω0kL∞ . If we now choose
δ = kω0k1/2L∞
Z t
0 ku(s)k2L∞ds −1/2
, we obtain the bound
ku(t)kL∞ ≤ C2ku0kL∞+ 2C2kω0k1/2L∞
Z t
0 ku(s)k2L∞ds 1/2
, (2.29)
which holds for any t∈ (0, T∗). Finally, squaring both sides of (2.29) and applying Gronwall’s lemma (see Section 4.4), we arrive at the inequality
ku(t)k2L∞ ≤ 2C22ku0k2L∞ exp
8C22kω0kL∞t
, t∈(0, T∗), (2.30) which shows that the normku(t)kL∞ cannot blow up in finite time. ThusT∗ =∞, and estimate
(2.30) holds for all t >0.
Remark 2.9. Theorem 1.1 is an immediate consequence of Propositions 2.4 and 2.7.
3 Uniformly local energy estimates
In the study of nonlinear partial differential equations on unbounded spatial domains, if one considers solutions that do not decay to zero at infinity, it is not always convenient to use function spaces based on the uniform norm k · k∞, because those spaces may not take into account some essential properties of the system, such as locally conserved or locally dissipated quantities. From this point of view, the larger family ofuniformly local Lebesgue spacesoffers an interesting compromise between simplicity and flexibility. In the analysis of evolution PDE’s, uniformly local spaces were introduced by T. Kato in 1975 [23], and subsequently used by many authors, see [3, 9, 12, 15, 21, 33, 34, 46] for a few examples. For the reader’s convenience, the definition and the main properties of those spaces are given in the Appendix, see Section 4.3. In short, if 1≤p <∞, the uniformly local Lebesgue space Lpul(Rd) is the completion of the space of all bounded and uniformly continuous functions on Rdwith respect to the norm
kfkLpul = sup
x∈Rd
Z
|y−x|≤1|f(y)|pdy
!1/p
. (3.1)
Our goal in this section is to control the solutions of the two-dimensional Navier-Stokes equations in the uniformly local energy spaceL2ul(R2). This can be done using uniformly local energy esti-mates, a flexible and powerful technique that we first explain on a simple example in Section 3.1 before applying it to the original problem in Section 3.2.
3.1 Uniformly local energy estimates for the heat equation
In this section we show on a simple example how uniformly local energy estimates can be used to obtain information on the solutions of partial differential equations on unbounded domains.
We concentrate on the linear heat equation on Rd, with nondecaying initial data u0. In that particular example, the solution can be written in explicit form, but we shall not use the heat kernel (4.22) because we want to develop robust methods that can be applied to more com-plicated situations, such as the two-dimensional Navier-Stokes equations which will considered later.
Let u0 ∈L2ul(Rd), and let u(x, t) be the solution of the heat equation
∂tu(x, t) = ∆u(x, t) , x∈Rd, t >0 , (3.2) with initial data u(·,0) =u0. It follows from Proposition 4.6 below that u ∈C0(R+, L2ul(Rd)), and our goal is to derive accurate bounds on u using localized energy estimates. Let ρ :Rd→ (0,+∞) be a Lipschitz continuous function satisfying the assumptions of Proposition 4.7 and such that|∇ρ(x)| ≤ρ(x) for almost every x∈Rd. Typical examples are
ρ(x) = e−|x| , or ρ(x) = 1
(m+|x|)m where m > d . (3.3) Note thatρ cannot decay to zero faster than an exponential as|x| → ∞, because of the assump-tion |∇ρ| ≤ρ. For anyR >0, we also defineρR(x) =ρ(x/R).
Since the solution u of (3.2) is smooth and bounded fort >0, we can compute 1 Using Gronwall’s lemma (see Section 4.4), we deduce that
Z for allt >0. This estimate looks rather pessimistic, because it predicts an exponential growth of the solution ast→ ∞, but one should keep in mind thatR
ρRu20dx <∞is the only assumption on the initial data that was really used in the derivation of (3.4). If ρ(x) = e−|x|, this means thatu0 is allowed to grow exponentially as|x| → ∞, in which case the solution of (3.2) indeed grows exponentially in time. we obtain from (3.4) and (3.5)
Z
for all t > 0. This estimate is clearly superior to (3.4), because the right-hand side grows only polynomially as t → ∞. Another elementary but important observation is that the same estimate holds if we replace ρR(x) with ρR(x−y) for any fixed y ∈Rd. Taking the supremum for all positive times, and (3.7) is sharp in that particular case, as far as the time dependence is concerned.
Unfortunately, the approach developed so far does not allow to bound the normku(t)kL2ul in an optimal way. Indeed, the best we can deduce directly from (3.6) is
ku(t)kL2ul ≤ Cku0kL2ul(1 +t)d/4 , t≥0 ,
which is not sharp in view of Proposition 4.6. To improve that result, a possibility is to use uniformly local Lp estimates for higher values ofp. Indeed, ifp∈N∗ we have as before
1
3.2 Uniformly local energy estimates for the 2D Navier-Stokes equations